The Unite union will learn on Thursday if its appeal against an injunction stopping planned strikes has succeeded.
It has sought to appeal against a legal decision to grant BA an injunction preventing industrial action.
Unite's joint general secretary Tony Woodley had called that decision "an absolute disgrace".
Three senior judges including the Lord Chief Justice have been hearing union arguments, and the reserved judgement will be handed out at 0930BST.
The BBC's Ben Ando said: "British Airways lawyers argued that because Unite had broken the rules by omitting them from the final declaration, any strike would be unlawful - and a judge agreed.
"The union has spent the day trying to convince the Court of Appeal that was wrong; and will find out on Thursday whether it's succeeded."
A successful union appeal could see strikes go ahead later in the month.
John Hendy QC, for Unite, told the court on Tuesday that if the appeal judges discharged the injunction, then industrial action would be "reinstated from an appropriate time thereafter".
But he added that should a settlement between BA and Unite be reached it would mean strikes not going ahead.
For BA, David Reade QC, had said it was the firm's case that there had been a "complete failure" by the union to discharge its obligations in relation to the communication of the information about the ballot result.
If Unite's legal efforts fail it says it will ballot members again on industrial action.
It would be the third such ballot since November last year, with the first two ballots declared invalid by the High Court on technicalities.
'Right to strike'
On Monday, a High Court judge ruled that Unite had failed to tell its members about the 11 spoilt ballot papers in the last strike vote.
Unite's joint general secretary Derek Simpson added that the union had done its utmost to tell its 12,000 members about the 11 ballot papers: "We posted it on the web, we put notices on our noticeboard and we distributed leaflets."
Brendan Barber, general secretary of the TUC, said he hoped an appeal hearing was granted, warning that a basic right to industrial action was at risk.
"Yesterday's judgement… sets a worrying precedent that calls in to question whether unions will have the right to organised lawful industrial action in the future," he said.
"The hurdles that are being placed in their way are now so complex that we're at risk of seeing important rights eroded."
Despite the ruling in favour of BA on Monday, flights have been disrupted with the airline unable to reinstate all flights.
BA says that at Heathrow, 90% of long-haul and 50% of short-haul flights are operating normally, meaning a passenger list of 60,000 people will fly from the airport, compared with a normal number for the time of year of 85,000.
The airline says passengers should not come to the airport unless they have a checked that their flight is unaffected.
BA says Gatwick and London City airports are unaffected.
The legal action by BA came after talks between the airline and Unite failed to reach agreement.
Although the two sides have agreed on the cost-cutting measures at the centre of the dispute, there is no agreement on the reinstatement of travel perks initially withdrawn from striking staff.
Unite has also called for more than 50 staff members who have been disciplined over their role in previous strikes to be reinstated.
Earlier, BA apologised to some customers whose travel plans would be affected.
BA boss Willie Walsh said the strike had been "unjustified" and he hoped the injunction would now give those involved in the dispute time to pause and reflect on the issues.
The airline and union had been in talks at Acas on Monday to try to find a way forward on disagreements over BA's cost-cutting plans.
But they came to an end when the court injunction decision became known.
In his ruling, Mr Justice McCombe said: "I am unable to say it is sufficiently clear that the union took the steps required by law at the time they were required."
And he said that the "balance of convenience" in his view required the granting of an injunction.
The first strike had been scheduled to begin on Tuesday, ending on 22 May, with the three further strikes planned to begin on 24 May, 30 May and 5 June.